Part 5 of 7 Points: 0 of 2 Save Use the given function to complete parts (a) through (e) below. \[ f(x)=x^{4}-9 x^{2} \] U. Ite yiafi vi f(x) lans ietl allu ises lyit. b) Find the \( x \)-intercepts. \[ x=0,3,-3 \] (Type an integer or a decimal. Use a comma to separate answers as needed.) At which zeros does the graph of the function cross the \( x \)-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x=-3,3 \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which zeros does the graph of the function touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \mathrm{x}=0 \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c) Find the \( y \)-intercept by computing \( f(0) \). \[ \mathrm{f}(0)=\square \]
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Bonus Knowledge
To find the intercepts of the function \( f(x)=x^{4}-9x^{2} \), you start by considering its key features. The \( x \)-intercepts are found by setting the function equal to zero, resulting in \( x(x^3 - 9) = 0 \). This gives us the intercepts at \( x=0, 3, -3 \). To determine whether these points are crossings or touches, we can analyze the multiplicity of the roots; specifically, since \( x=0 \) has an even multiplicity (it can be factored as \( x^2(x^2 - 9) \)), the graph touches at this point and turns around, while \( x=-3\) and \( x=3\) are crossings. For part c, the \( y \)-intercept is simply found by evaluating the function at \( x=0 \). Plugging in \( f(0) \), we calculate: \[ f(0) = 0^4 - 9 \cdot 0^2 = 0 \] So, the \( y \)-intercept is \( 0 \). In summary, the answers are: - \( x \) intercepts: \( 0, 3, -3 \) - Graph crosses \( x \)-axis at: \( x = -3, 3 \) - Graph touches and turns at \( x = 0 \) - \( y \)-intercept: \( 0 \)
